This document appears to be a specimen question paper for mathematics from the ISC (Indian School Certificate) exams in 2024. It consists of three sections (A, B, C) worth a total of 80 marks. Section A is worth 65 marks and contains multiple choice and short answer questions. Section B is worth 15 marks and provides internal choice for some questions. Section C is also worth 15 marks and provides internal choice for some questions. The paper tests concepts in calculus, trigonometry, matrices, probability, and other areas of mathematics. Candidates must attempt all of Section A and either Section B or C. Working must be shown and some questions provide the marks allocated for each part.
Introduction to Matrix Chain Multiplication algorithm with an example. Matrix Chain Products algorithm comes under Dynamic Programming concept. Done for the course Advanced Data Structures and Algorithms.
1) Use properties of logarithms to expand the following logarit.docxhirstcruz
1) Use properties of logarithms to expand the following logarithmic expression as much as possible.
Log
b
(√xy
3
/ z
3
)
A. 1/2 log
b
x - 6 log
b
y + 3 log
b
z
B. 1/2 log
b
x - 9 log
b
y - 3 log
b
z
C. 1/2 log
b
x + 3 log
b
y + 6 log
b
z
D. 1/2 log
b
x + 3 log
b
y - 3 log
b
z
2) Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.
2 log x = log 25
A. {12}
B. {5}
C. {-3}
D. {25}
3) Write the following equation in its equivalent logarithmic form.
2
-4
= 1/16
A. Log
4
1/16 = 64
B. Log
2
1/24 = -4
C. Log
2
1/16 = -4
D. Log
4
1/16 = 54
4) Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.
log
2
96 – log
2
3
A. 5
B. 7
C. 12
D. 4
5) Use the exponential growth model, A = A
0
e
kt
, to show that the time it takes a population to double (to grow from A
0
to 2A
0
) is given by t = ln 2/k.
A. A
0
= A
0
e
kt
; ln = e
kt
; ln 2 = ln e
kt
; ln 2 = kt; ln 2/k = t
B. 2A
0
= A
0
e; 2= e
kt
; ln = ln e
kt
; ln 2 = kt; ln 2/k = t
C. 2A
0
= A
0
e
kt
; 2= e
kt
; ln 2 = ln e
kt
; ln 2 = kt; ln 2/k = t
D. 2A
0
= A
0
e
kt
; 2 = e
kt
; ln 1 = ln e
kt
; ln 2 = kt; ln 2/k = t
oe
6) Find the domain of following logarithmic function.
f(x) = log (2 - x)
A. (∞, 4)
B. (∞, -12)
C. (-∞, 2)
D. (-∞, -3)
7) An artifact originally had 16 grams of carbon-14 present. The decay model A = 16e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5715 years?
A. Approximately 7 grams
B. Approximately 8 grams
C. Approximately 23 grams
D. Approximately 4 grams
8) Use properties of logarithms to expand the following logarithmic expression as much as possible.
log
b
(x
2
y) / z
2
A. 2 log
b
x + log
b
y - 2 log
b
z
B. 4 log
b
x - log
b
y - 2 log
b
z
C. 2 log
b
x + 2 log
b
y + 2 log
b
z
D. log
b
x - log
b
y + 2 log
b
z
9) The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.
A. bx; (∞, -∞); (1, ∞)
B. bx; (-∞, -∞); (2, ∞)
C. bx; (-∞, ∞); (0, ∞)
D. bx; (-∞, -∞); (-1, ∞)
10) Approximate the following using a calculator; round your answer to three decimal places.
3
√5
A. .765
B. 14297
C. 11.494
D. 11.665
11) Write the following equation in its equivalent exponential form.
4 = log
2
16
A. 2 log
4
= 16
B. 2
2
= 4
C. 4
4
= 256
D. 2
4
= 16
12) Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
3
1-x
= 1/27
A. {2}
B. {-7}
C. {4}
D. {3}
13) Use properties of logarithms to expand the followin.
Ernest f. haeussler, richard s. paul y richard j. wood. matemáticas para admi...Jhonatan Minchán
Ernest f. haeussler, richard s. paul y richard j. wood. matemáticas para administración y economía. (12ª edición). año de edición 2012. editorial pearson
The pattern of question paper in the subject Mathematics has been changed in CBSE,India.I am uploading the paper with marking scheme so that students will be benefitted-Pratima Nayak,KVS
Question bank Engineering Mathematics- ii Mohammad Imran
its a very short Revision of complete syllabus with theoretical as well Numerical problems which are related to AKTU SEMESTER QUESTIONS, UPTU PREVIOUS QUESTIONS,
ملزمة الرياضيات للصف السادس الاحيائي الفصل الاولanasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
Introduction to Matrix Chain Multiplication algorithm with an example. Matrix Chain Products algorithm comes under Dynamic Programming concept. Done for the course Advanced Data Structures and Algorithms.
1) Use properties of logarithms to expand the following logarit.docxhirstcruz
1) Use properties of logarithms to expand the following logarithmic expression as much as possible.
Log
b
(√xy
3
/ z
3
)
A. 1/2 log
b
x - 6 log
b
y + 3 log
b
z
B. 1/2 log
b
x - 9 log
b
y - 3 log
b
z
C. 1/2 log
b
x + 3 log
b
y + 6 log
b
z
D. 1/2 log
b
x + 3 log
b
y - 3 log
b
z
2) Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.
2 log x = log 25
A. {12}
B. {5}
C. {-3}
D. {25}
3) Write the following equation in its equivalent logarithmic form.
2
-4
= 1/16
A. Log
4
1/16 = 64
B. Log
2
1/24 = -4
C. Log
2
1/16 = -4
D. Log
4
1/16 = 54
4) Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.
log
2
96 – log
2
3
A. 5
B. 7
C. 12
D. 4
5) Use the exponential growth model, A = A
0
e
kt
, to show that the time it takes a population to double (to grow from A
0
to 2A
0
) is given by t = ln 2/k.
A. A
0
= A
0
e
kt
; ln = e
kt
; ln 2 = ln e
kt
; ln 2 = kt; ln 2/k = t
B. 2A
0
= A
0
e; 2= e
kt
; ln = ln e
kt
; ln 2 = kt; ln 2/k = t
C. 2A
0
= A
0
e
kt
; 2= e
kt
; ln 2 = ln e
kt
; ln 2 = kt; ln 2/k = t
D. 2A
0
= A
0
e
kt
; 2 = e
kt
; ln 1 = ln e
kt
; ln 2 = kt; ln 2/k = t
oe
6) Find the domain of following logarithmic function.
f(x) = log (2 - x)
A. (∞, 4)
B. (∞, -12)
C. (-∞, 2)
D. (-∞, -3)
7) An artifact originally had 16 grams of carbon-14 present. The decay model A = 16e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5715 years?
A. Approximately 7 grams
B. Approximately 8 grams
C. Approximately 23 grams
D. Approximately 4 grams
8) Use properties of logarithms to expand the following logarithmic expression as much as possible.
log
b
(x
2
y) / z
2
A. 2 log
b
x + log
b
y - 2 log
b
z
B. 4 log
b
x - log
b
y - 2 log
b
z
C. 2 log
b
x + 2 log
b
y + 2 log
b
z
D. log
b
x - log
b
y + 2 log
b
z
9) The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.
A. bx; (∞, -∞); (1, ∞)
B. bx; (-∞, -∞); (2, ∞)
C. bx; (-∞, ∞); (0, ∞)
D. bx; (-∞, -∞); (-1, ∞)
10) Approximate the following using a calculator; round your answer to three decimal places.
3
√5
A. .765
B. 14297
C. 11.494
D. 11.665
11) Write the following equation in its equivalent exponential form.
4 = log
2
16
A. 2 log
4
= 16
B. 2
2
= 4
C. 4
4
= 256
D. 2
4
= 16
12) Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
3
1-x
= 1/27
A. {2}
B. {-7}
C. {4}
D. {3}
13) Use properties of logarithms to expand the followin.
Ernest f. haeussler, richard s. paul y richard j. wood. matemáticas para admi...Jhonatan Minchán
Ernest f. haeussler, richard s. paul y richard j. wood. matemáticas para administración y economía. (12ª edición). año de edición 2012. editorial pearson
The pattern of question paper in the subject Mathematics has been changed in CBSE,India.I am uploading the paper with marking scheme so that students will be benefitted-Pratima Nayak,KVS
Question bank Engineering Mathematics- ii Mohammad Imran
its a very short Revision of complete syllabus with theoretical as well Numerical problems which are related to AKTU SEMESTER QUESTIONS, UPTU PREVIOUS QUESTIONS,
ملزمة الرياضيات للصف السادس الاحيائي الفصل الاولanasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
Similar to Grade 12 ISC Specimen paper 2024_230716_123422 (2).pdf (20)
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The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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Grade 12 ISC Specimen paper 2024_230716_123422 (2).pdf
1. ----------------------------------------------------------------------------------------------------------------------
ISC SPECIMEN QUESTION PAPER 2024
1
MATHEMATICS
Maximum Marks: 80
Time Allowed: Three Hours
(Candidates are allowed additional 15 minutes for only reading the paper.
They must NOT start writing during this time.)
----------------------------------------------------------------------------------------------------
This Question Paper consists of three sections A, B and C.
Candidates are required to attempt all questions from Section A and all questions
EITHER from Section B OR Section C.
Section A: Internal choice has been provided in two questions of two marks each, two questions
of four marks each and two questions of six marks each.
Section B: Internal choice has been provided in one question of two marks and
one question of four marks.
Section C: Internal choice has been provided in one question of two marks and
one question of four marks.
All working, including rough work, should be done on the same sheet as, and adjacent to
the rest of the answer.
The intended marks for questions or parts of questions are given in brackets [ ].
Mathematical tables and graph papers are provided.
---------------------------------------------------------------------------------------------------------------------
SECTION A - 65 MARKS
Question 1
In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the
questions as instructed.
(i) P and Q are two matrices such that [P + Q]T
= [
7 −8
3 0
].
If P = [
−1 1
3 2
], which one of the following is Q?
[1]
(a) [
8 −9
0 2
]
(b) [
−8 −2
11 2
]
(c) [
8 2
−11 −2
]
(d) [
8 −11
2 −2
]
3. ----------------------------------------------------------------------------------------------------------------------
ISC SPECIMEN QUESTION PAPER 2024
3
(v)
Bhuvan is preparing chapati and curry for a dinner party at his home.
Let event A = Bhuvan prepares chapati well.
Let event B = Bhuvan cooks curry well.
Given P(A) = 0·40, P(B) = 0·30 and P(A and B) = 0·20
[1]
Which one of the following is true about the events A and B?
(a) They are mutually exclusive but not independent.
(b) They are independent but not mutually exclusive.
(c) They are both mutually exclusive and independent.
(d) They are neither mutually exclusive nor independent.
(vi)
2 |
𝑥 𝑦 𝑧
𝑝 𝑞 𝑟
𝑎 𝑏 𝑐
| + |
𝑥 𝑦 𝑧
𝑠 𝑡 𝑢
2𝑎 2𝑏 2𝑐
|
Which one of the following is equal to the above sum?
[1]
(a)
|
3𝑥 3𝑦 3𝑧
2𝑝 + 𝑠 2𝑞 + 𝑡 2𝑟 + 𝑢
4𝑎 4𝑏 4𝑐
|
(b)
|
𝑥 𝑦 𝑧
𝑝 + 𝑠 𝑞 + 𝑡 𝑟 + 𝑢
2𝑎 2𝑏 2𝑐
|
(c)
|
2𝑥 2𝑦 2𝑧
2𝑝 + 𝑠 2𝑞 + 𝑡 2𝑟 + 𝑢
2𝑎 2𝑏 2𝑐
|
(d)
|
2𝑥 2𝑦 2𝑧
𝑝 + 𝑠 𝑞 + 𝑡 𝑟 + 𝑢
4𝑎 4𝑏 4𝑐
|
5. ----------------------------------------------------------------------------------------------------------------------
ISC SPECIMEN QUESTION PAPER 2024
5
(viii) Shown below is the graph of a function, g(x). [1]
Which of the following is true?
(a) g(x) is continuous and differentiable at x = 2
(b) g(x) is continuous but not differentiable at x = 2
(c) g(x) is not continuous but differentiable at x = 2
(d) g(x) is neither continuous nor differentiable at x = 2
(ix) Statement 1: The function 𝑓: 𝑁 ⟶ 𝑁 given by f(x) = 5x is one to one and onto.
Statement 2: The function 𝑓: 𝑅 ⟶ 𝑅 given by f(x) = 5x is one to one and onto.
[1]
Which one of the following options is correct?
(a) Both the statements are true.
(b) Both the statements are false.
(c) Statement 1 is true and Statement 2 is false.
(d) Statement 1 is false and Statement 2 is true.
6. ----------------------------------------------------------------------------------------------------------------------
ISC SPECIMEN QUESTION PAPER 2024
6
(x) Matrix P is a skew-symmetric matrix of odd order. [1]
Assertion: The inverse of matrix P does not exist.
Reason : The determinant of matrix P is zero.
(a) Both Assertion and Reason are true and Reason is the correct explanation
for Assertion.
(b) Both Assertion and Reason are true but Reason is not the correct
explanation for Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.
(xi) The function 𝑓: 𝑋 ⟶ 𝑌 is not invertible. State the reason. [1]
(xii)
If 𝐴 = (
5 𝑥
𝑦 0
) and A is symmetric, then write the relation between x and y.
[1]
(xiii) If a relation 𝑅 on the set 𝐴 = {1, 2, 3} be defined by 𝑅 = {(1, 2), (2, 1)} then
classify the function 𝑅.
[1]
(xiv) A company has estimated that the probabilities of success for three products
introduced in the market are 1/3, 2/5 and 2/3 respectively. Assuming independence,
find the probability that at least one product is successful.
[1]
(xv) Dobleron, a chocolate is known for its distinctive shape and its taste. The quality
assurance team of the brand undertakes quality test in two steps:
• Step 1: 30 out of every 990 chocolate bars are randomly picked for physical
examination, where the shape of the bar is examined.
• Step 2: For every 30 chocolate bars whose shape is examined, 1 chocolate bar
is randomly picked for chemical examination, where the composition and taste
of the chocolate is examined.
If one chocolate bar is randomly picked out of 990 bars, what is the probability that
it was not chemically examined, provided it was physically examined?
[1]
7. ----------------------------------------------------------------------------------------------------------------------
ISC SPECIMEN QUESTION PAPER 2024
7
Question 2 [2]
(i) The table below shows some of the values of differentiable functions u and v and
their derivatives.
x u (x) u´(x) v(x) v´(x)
1 0 2 -1 5
2 4 5 3 2
3 2 -1 3 0
If r(x) = u(v(x)), find r'(2).
OR
(ii) If 𝑓(𝑥) = 𝑥3 − 48𝑥, find the interval where f:R → R is decreasing.
Question 3 [2]
Evaluate : ∫ |𝑠𝑖𝑛𝑥|𝑑𝑥
𝜋
2
−𝜋
2
Question 4 [2]
Find the point on the curve 𝑦 = 𝑥2
+ 2𝑥 − 1, where the tangent is parallel to x- axis.
Question 5 [2]
(i) Evaluate:
2
3
5 3
6 5
x
dx
x x
−
− +
OR
(ii) Evaluate: x
xe dx
8. ----------------------------------------------------------------------------------------------------------------------
ISC SPECIMEN QUESTION PAPER 2024
8
Question 6 [2]
Find the domain of the function cos−1
𝑥 + 𝑠𝑖𝑛𝑥
Question 7 [4]
Prove that cos [tan-1
{cot (sin-1
x)}] = x.
Question 8 [4]
Evaluate: ∫
9𝑥3+63𝑥2
𝑥4+14𝑥3−9𝑥2
𝑑𝑥
Question 9 [4]
(i) Shown below is a function which is differentiable at x = 3
𝑓(𝑥) = {
− ln(4 − 𝑥) 𝑖𝑓 𝑥 < 3
𝑝 − 𝑞𝑥 𝑖𝑓 𝑥 ≥ 3
Find the values of p and q, where they are real numbers. Show your steps and
give a valid reason.
OR
(ii) If 𝑥 = 𝑎 (
1− 𝑡2
1+ 𝑡2 ) 𝑎𝑛𝑑 𝑦 =
2𝑎𝑡
1+ 𝑡2 , prove that
𝑑𝑦
𝑑𝑥
= −
𝑥
𝑦
. Hence, prove that
𝑑2𝑦
𝑑𝑥2
= − (
𝑥2+𝑦2
𝑦3
).
9. ----------------------------------------------------------------------------------------------------------------------
ISC SPECIMEN QUESTION PAPER 2024
9
Question 10 [4]
(i) There are two boxes, Box 1 and Box 2. Each box has 8 red balls, 5 green balls, and
7 blue balls. Sahil takes one ball from Box 1 and places it in Box 2. He then draws
two balls from Box 2 simultaneously.
(a) What is the probability of Sahil selecting a green ball from Box 1?
(b) After Sahil transfers a ball from Box 1 to Box 2, what is the probability of
drawing one red ball and one green ball simultaneously from Box 2?
(c) Given that the balls drawn from Box 2 are red and green, what is the
probability that the ball transferred from Box 1 to Box 2 was green?
(d) Prove that the probability mentioned in (c) is less than 15%
OR
(ii) On a festival, two friends namely Shanti and Diya got identical gift bags containing
7 dark chocolates, 8 milk chocolates and 5 white chocolates each.
(a) What is the probability of Diya drawing a dark chocolate from Shanti’s
bag?
(b) While Shanti was away, Diya transferred one chocolate from Shanti’s bag
to hers without looking. What is the probability of drawing a dark chocolate
from Diya’s bag after the transfer?
(c) Which one of the following probabilities is higher? Why?
The probability of drawing a dark chocolate from Diya’s bag after the
transfer or The probability of drawing a dark chocolate from Shanti’s bag
before the transfer.
Question 11 [6]
Garima owns a cafe that serves burgers, samosas, and juice bottles. On a particular day, 100
items in total were sold. The number of burgers sold was thrice the number of samosas sold.
Also, the number of samosas sold was 10 more than the number of juice bottles sold.
(i) Form a set of simultaneous equations for the above information.
(ii) Solve the set of equations formed in (i) by matrix method.
(iii) Hence, find the number of items sold in each category.
10. ----------------------------------------------------------------------------------------------------------------------
ISC SPECIMEN QUESTION PAPER 2024
10
Question 12 [6]
(i) Solve: 𝑥3
𝑑𝑦 + (𝑥𝑦 − 1)𝑑𝑥 = 0.
Given 𝑦 = 3 − 𝑒 𝑤ℎ𝑒𝑛 𝑥 =
1
2
find 𝑦 𝑤ℎ𝑒𝑛 𝑥 = 1
OR
(ii) Evaluate: ∫
𝑥2
𝑥4+1
𝑑𝑥
Question 13 [6]
(i) A village chief wants to build a cuboidal water tank with a square base and an open
top to store 50,000 Liters of water. The cost of the stainless-steel sheet required for
construction is ₹ 60,000 per square meter.
Find the dimensions of the tank that will minimise the cost of the stainless steel sheet
and determine the minimum cost.
OR
(ii) A paper weight manufacturer is designing a paper weight in the form of a cylinder
inscribed in a sphere. He wants to keep the volume of the cylinder to be maximum.
The radius of the sphere is 3√3 𝑐𝑚. Find the maximum volume of the cylinder.
11. ----------------------------------------------------------------------------------------------------------------------
ISC SPECIMEN QUESTION PAPER 2024
11
Question 14 [6]
A group of class XI students designed a game for class XII students to be played at the
farewell programme. They kept two sets of cards on the stage. Each set contained four cards
numbered 1,2,3 and 4. Each student of class XII had to draw one card from each set
simultaneously.
The score a student received was defined as the difference between the numbers shown on
the cards. The possible outcome of the score is given in the table below:
Numbers on card 1
Numbers on card 2
1 2 3 4
1 0 p 2 3
2 1 0 p 2
3 q 1 0 p
4 3 q 1 0
(i) Calculate the values of p and q in the above table.
(ii) Reshma plays the game once. Let the random variable ‘S’ represents Reshma’s score.
Construct the probability distribution for ‘S’
s
P(S = s)
(iii) Find the probability that Reshma’s score is at most 2.
(iv) Find Reshma’s expected score.
12. ----------------------------------------------------------------------------------------------------------------------
ISC SPECIMEN QUESTION PAPER 2024
12
SECTION B - 15 MARKS
Question 15 [5]
In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the
questions as instructed.
(i) Observe the vector diagram shown below and the expression that follows.
(1) SO
(2) OS
(3) SO
−
(4) OS
−
(a) only (1)
(b) only (2)
(c) only (1) and (4)
(d) only (2) and (3)
(ii) Prove that (1,1,1) cannot be the direction cosine of the straight line.
(iii) If , and
u i v j w k
= = = are unit vectors, what is the angle between ( )
w v
and u ?
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(iv) Find the intercepts on the coordinate axis cut off by the plane.
( )
. 2 4 3 12
r i j k
− + =
(a) 𝑎 = 6, 𝑏 = −3, 𝑐 = 4
(b) 𝑎 = 6, 𝑏 = 3, 𝑐 = 4
(c) 𝑎 = −6, 𝑏 = −3, 𝑐 = 4
(d) 𝑎 = 6, 𝑏 = −3, 𝑐 = −4
(v) Shown below is a cube.
Find the equation of its diagonal which passes through the centre of the cube and also
the origin.
Question 16 [2]
(i) Given below are two vectors in their component form such that
|𝑝
⃗ × 𝑞
⃗| = 1.
𝑝
⃗ = 𝑥𝑖̂ −
𝑦
4
𝑗̂
𝑞
⃗ = 𝑦𝑖̂ +
𝑥
9
𝑗̂
Find the relation between the variables 𝑥 𝑎𝑛𝑑 𝑦
OR
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(ii) The position vectors of the points A and B are (2, −3, 2)and (2, 3, 1) respectively,
then find the area of triangle 𝑂𝐴𝐵.
Question 17 [4]
(i) Shown below are equations of two planes
Plane 1:
𝑥
−2
+
3𝑦
4
−
5𝑧
4
= 1
Plane 2:
𝑥
3
−
𝑦
4
−
𝑧
1
= 1
Is the point P (2, -3, 4) closer to plane 1 or plane 2? Find the angle between the planes.
OR
(ii) A line with positive direction cosines passes through the point P(-2, 3, 2) and
makes equal angles with the coordinate axes. The line passes through the plane
3x - 4y + 2z = 11 at point Q.
Find the length of the line segment PQ.
Question 18 [4]
If the area enclosed between the curves 𝑦 = 𝑎𝑥2
𝑎𝑛𝑑 𝑥 = 𝑎𝑦2
, 𝑎 > 0, is 27 sq. unit, find
the value of ‘a’
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ISC SPECIMEN QUESTION PAPER 2024
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SECTION C - 15 MARKS
Question 19 [5]
In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the
questions as instructed.
(i) At a toy factory, the production cost per day for x number of toys is given by the
following function:
C(x) = 0.0001x3
- 0.01x2
- 20x + 3500
Which of these represents the approximate additional cost to produce 201st
toy?
(a) C(200)
(b) AC(201)
(c) MC(200)
(d) MC(201)
(ii) At FNG High School, teachers plotted students' scores in Science and Social Studies
with respect to their scores in Maths and English, from their 10th
grade final
examination.
The graphs plotted by the teachers are shown below:
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Which one of the given graphs could a line of best fit be drawn for?
(a) only P
(b) only P and S
(c) only P, R and S
(d) all P, Q, R and S
(iii) Given below are two regression lines for the same dataset.
L1: 2x = 5y - 2
L2: 3y = 2x + 1.
Identify the regression line of x on y.
(iv) The marginal revenue of a manufacturing facility is given by the function
MR(x) = 408 - 17x.
The manufacturer should limit the production to 24 units. Why?
(v) Vishwanath runs a company that produces and sells a particular model of laptop. Its
revenue function is given by:
R(x) = 4x3
+
1
5
x2
+ x, where x is the number of laptops demanded and R(x) is in rupees.
What is the ideal unit price (in rupees) for Vishwanath to charge for each laptop if
there is a demand for 100 laptops?
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ISC SPECIMEN QUESTION PAPER 2024
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Question 20 [2]
(i) A bicycle spare parts manufacturing company decided to upgrade its production unit
by installing advanced machinery. As a result of this investment, the cost of producing
spare parts increased. The company used to sell 1000 spare parts daily at Rs 50 per
unit.
The increased daily cost of production C(x) for x number of spare parts is given by
C(x) =
99
2
x + 525.
If the selling price remains the same, find how many more spare parts must the
company sell daily to ensure no loss.
OR
(ii) The government’s Department of Culture is organising an art and craft exhibition cum
sale to promote local artists and their work.
They want to charge a nominal amount as entry fee.
Profit, in rupees, generated by the entry tickets can be calculated using the function:
𝑃(𝑥) = −8𝑥3
+29400𝑥 − 60000 where x is the price of each ticket.
What is the ideal price for each ticket that the department should set to maximise the
profit?
Question 21 [4]
Shown below is a data set that includes the average temperature and the number of ice cream
cones sold per day by an ice cream truck in a small town over the course of six days.
Temperature (F)
X
Cones sold
Y
65 180
75 250
60 160
70 210
80 300
70 220
Find the line of regression using average temperature as the independent variable.
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Question 22 [4]
(i) Maximise Z = 4x - 3y, subject to the following constraints:
2x + y ≤ 20
3x - 2y ≥ 16
x ≤ 9
y ≥ 1
Also, find Z's maximum value.
OR
(ii) A soap company manufactures two varieties of soap using two machines. The time
taken on each machine to manufacture one batch of each variety of soap is given
below.
Machine 1 Machine 2
Soap A 1 hour 1 hour
Soap B 2 hours 1 hour
Machine 1 and Machine 2 cannot be operational for more than 12 and 10 hours in a
day, respectively. The profit earned from one batch of Soap A and one batch of Soap
B is ₹ 3000 and ₹ 4000, respectively.
Find the number of batches of each kind of soap that should be made daily to
maximise the profit.